Chapter 10. Options Engineering with Applications
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of xT at time T > t. We call f (xT ) a payoff function. The functional form of f (.) is known if
the contract is well deﬁned.2 It is customary in textbooks to represent the pair {f (xT ), xT } as
in Figures 10-1a or 10-1b. Note that, here, we have a nonlinear upward sloping payoff function
that depends on the values assumed by xT only. The payoff diagram in Figure 10-1a is drawn
in a completely arbitrary fashion, yet, it illustrates some of the general principles of ﬁnancial
exposures. Let us review these.
First of all, for fairly priced exposures that have zero value of initiation, net exposures to a
risk factor, xT , must be negative for some values of the underlying risk. Otherwise, we would be
Payoff
f (xT )
A nonlinear exposure
xB
0
x0
xA
xT
xC
FIGURE 10-1a
Payoff
A linear exposure
xT
0
x0
Current value
FIGURE 10-1b
2 Here x can be visualized as a kxl vector of risk factors. To simplify the discussion, we will proceed as if there
t
is a single risk factor, and we assume that xt is a scalar random variable.
1. Introduction
279
making positive gains, and there would be no risk of losing money. This would be an arbitrage
opportunity. Swap-type instruments fall into this category. If, on the other hand, the ﬁnal payoffs
of the contract are nonnegative for all values of xT , the exposure has a positive value at initiation,
and to take the position an upfront payment will have to be made. Option positions have this
characteristic.3
Second, exposures can be convex, concave, or linear with respect to xT , and this has relevance for an investor or market professional. The implication of linearity is obvious: the sensitivity of the position to movements in xT is constant. The relevance of convexity was discussed
in Chapters 8 and 9. With convexity, movements in volatility need to be priced in, and again
options are an important category here.
Finally, it is preferable that the payoff functions f (xT ) depend only on the underlying risk,
xT , and do not move due to extraneous risks. We saw in Chapters 8 and 9 that volatility positions
taken with options may not satisfy this requirement. The issue will be discussed in Chapter 14.
1.1.1.
Examples of xt
The discussion thus far dealt with an abstract underlying, xt . This underlying can be almost any
risk the human mind can think of. The following lists some well-known examples of xt .
• Various interest rates. The best examples are Libor rates and swap rates. But the commercial paper (CP) rate, the federal funds rate, the index of overnight interest rates (an
example of which is EONIA, Euro Over Night Index Average), and many others are also
used as reference rates.
• Exchange rates, especially major exchange rates such as dollar-euro, dollar-yen, dollarsterling (“cable”), and dollar-Swiss franc.
• Equity indices. Here also the examples are numerous. Besides the well-known U.S.
indices such as the Dow, Nasdaq, and the S&P500, there are European indices such as
CAC40, DAX, and FTSE100, as well as various Asian indices such as the Nikkei 225
and emerging market indices.
• Commodities are also quite amenable to such positions. Futures on coffee, soybeans,
and energy are other examples for xT .
• Bond price indices. One example is the EMBI + prepared by JPMorgan to track emerging
market bonds.
Besides these well-known risks, there are more complicated underlyings that, nevertheless,
are central elements in ﬁnancial market activity:
1. The underlying to the option positions discussed in this chapter can represent volatility
or variance. If we let the percentage volatility of a price, at time t, be denoted by σt , then
the time T value of the underlying xT may be deﬁned as
T
xT =
t
σu2 Su2 du
(1)
where St may be any risk factor. In this case, xT represents the total variance of St during
the interval [t, T ]. Volatility is the square root of xT .
2. The correlation between two risk factors can be traded in a similar way.
3. The underlying, xt , can also represent the default probability associated with a counterparty or instrument. This arises in the case of credit instruments.
3
The market maker will borrow the needed funds and buy the option. Position will still have zero value at initiation.
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4. The underlying can represent the probability of an extraordinary event happening. This
would create a “Cat” instrument that can be used to buy insurance against various
catastrophic events.
5. The underlying, xt , can also be a nonstorable item such as electricity, weather, or
bandwidth.
Readers who are interested in the details of such contracts or markets should consult Hull (2008).
In this chapter, we limit our attention to the engineering aspects of option contracts.
2.
Option Strategies
We divide the engineering of option strategies into two broad categories. First, we consider
the classical option-related methods. These will cover strategies used by market makers as
well as retail investors. They will themselves be divided into two groups, those that can be
labeled directional strategies, and those that relate to views on the volatility of some underlying
instrument. The second category involves exotic options, which we consider as more efﬁcient
and sometimes cheaper alternatives to the classical option strategies. The underlying risks can
be any of those mentioned in the previous section.
2.1. Synthetic Long and Short Positions
We begin with strategies that utilize options essentially as directional instruments, starting with
the creation of long and short positions on an asset. Options can be used to create these positions
synthetically.
Consider two plain vanilla options written on a forward price Ft of a certain asset. The ﬁrst
is a short put, and the second a long call, with prices P (t) and C(t) respectively, as shown in
Figure 10-2. The options have the same strike price K, and the same expiration time T .4 Assume
that the Black-Scholes conditions hold, and that both options are of European style. Importantly,
the underlying asset does not have any payouts during [t, T ]. Also, suppose the appropriate short
rate to discount future cash ﬂows is constant at r.
Now consider the portfolio
{1 Long K-Call, 1 Short K-Put}
(2)
At expiration, the payoff from this portfolio will be the vertical sum of the graphs in Figure 10-2
and is as shown in Figure 10-3. This looks like the payoff function of a long forward contract
entered into at K. If the options were at-the-money (ATM) at time t, the portfolio would exactly
duplicate the long forward position and hence would be an exact synthetic. But there is a
close connection between this portfolio and the forward contract, even when the options are
not ATM.
At expiration time T , the value of the portfolio is
C(T ) − P (T ) = FT − K
(3)
where FT is the time-T value of the forward price. This equation is valid because at T , only one
of the two options can be in-the-money. Either the call option has a value of FT − K while the
other is worthless, or the put is in-the-money and the call is worthless, as shown in Figure 10-2.
4
Short calls and long puts lead to symmetric results and are not treated here.
2. Option Strategies
Gain
Payoff from
long K-call
Call is worth
(FT 2 K ) at expiration
0
FT
K
FT
Loss
Gain
Payoff from
short K-put
K
0
FT
Expiration FT
Put expires worthless here
Loss
FIGURE 10-2
1
0
K
Joint payoff
(long call, short put)
2
FIGURE 10-3
FT
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Subtract the time-t forward price, Ft , from both sides of this equation to obtain
C(T ) − P (T ) + (K − Ft ) = FT − Ft
(4)
This expression says that the sum of the payoffs of the long call and the short put plus (K − Ft )
units of cash should equal the time-T gain or loss on a forward contract entered into at Ft , at
time t.
Take the expectation of equation (4). Then the time t value of the portfolio,
{1 Long K-Call, 1 Short K-Put, e−r(T −t) (K − Ft ) Dollars}
(5)
should be zero at t, since credit risks and the cash ﬂows generated by the forward and the
replicating portfolio are the same. This implies that
C(t) − P (t) = e−r(T −t) (Ft − K)
(6)
This relationship is called put-call parity. It holds for European options. It can be expressed in
terms of the spot price, St , as well. Assuming zero storage costs, and no convenience yield:5
Ft = er(T −t) St
(7)
Substituting in the preceding equation gives
C(t) − P (t) = (St − e−r(T −t) K)
(8)
Put-call parity can thus be regarded as another result of the application of contractual equations,
where options and cash are used to create a synthetic for the St . This situation is shown in
Figure 10-4.
Slope = 1
1
Long call
K
(FT − K)
FT
0
St
FT
Short put
2
FIGURE 10-4
5 Here the r is the borrowing cost and, as discussed in Chapter 4, is a determinant of forward prices. The convenience
yield is the opposite of carry cost. Some stored cash goods may provide such convenience yield and affect Ft .
2. Option Strategies
2.1.1.
283
An Application
Option market makers routinely use the put-call parity in exploiting windows of arbitrage
opportunities. Using options, market makers construct synthetic futures positions and then trade
them against futures contracts. This way, small and temporary differences between the synthetic
and the true contract are converted into “riskless” proﬁts. In this section we discuss an example.
Suppose, without any loss of generality, that a stock is trading at
St = 100
(9)
and that the market maker can buy and sell at-the-money options that expire in 30 days. Suppose
also that the market maker faces a funding cost of 5%. The stock never pays dividends and there
are no corporate actions.
Also, and this is the real-life part, the market maker faces a transaction cost of 20 cents per
traded option and a transaction cost of 5 cents per traded stock. Finally, the market maker has
calculated that to be able to continue operating, he or she needs a margin of .25 cent per position.
Then, we can apply put-call parity and follow the conversion strategy displayed in Figure 10-5.
Borrow necessary funds overnight for 30 days, and buy the stock at price St . At the same
time, sell the St -call and buy the St -put that expires in 30 days, to obtain the position
shown in Figure 10-5.
The position is fully hedged, as any potential gains due to movement in St will cover the
potential losses. This means that the only factors that matter are the transaction costs and any
price differentials that may exist between the call and the put. The market maker will monitor the difference between the put and call premiums and take the arbitrage position shown in
Figure 10-5 if this difference is bigger than the total cost of the conversion.
Example:
Suppose St = 100, and 90-day call and put options trade actively. The interest cost is 5%.
A market maker has determined that the call premium, C(t), exceeds the put premium,
P (t), by $2.10:
C(t) − P (t) = 2.10
(10)
The stock will be purchased using borrowed funds for 90 days, and the ATM put is
purchased and held until expiration, while the ATM call is sold. This implies a funding
cost of
100(.05)
90
360
= $1.25
Add all the costs of the conversion strategy:
Cost per security
$
Funding cost
Stock purchase
Put purchase
Call sale
Operating costs
1.25
.05
.20
.20
.25
Total cost
1.95
(11)
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Stock
1
Long stock position
funded with money
market loan
0
St
100 5 St
2
1
0
St
K 5 100
2
1
Short ATM call
Long ATM put
St
0
K 5 100
2
Adding together . . . If prices are different “enough”
then there will be arbitrage opportunity.
1
Stock funded with loan
0
St
100
2
Call 1 Put position
FIGURE 10-5
2. Option Strategies
285
The market maker incurs a total cost of $1.95. It turns out that under these conditions,
the net cash position will be positive:
Net proﬁt = 2.10 − 1.95
(12)
and the position is worth taking.
If, in the example just discussed, the put-call premium difference is negative, then the market
maker can take the opposite position, which would be called a reversal.6
2.1.2.
Arbitrage Opportunity?
An outside observer may be surprised to hear that such “arbitrage” opportunities exist, and that
they are closely monitored by market makers on the trading ﬂoor. Yet, such opportunities are
available only to market makers on the “ﬂoor” and may not even constitute arbitrage in the usual
sense.
This is because of the following: (1) Off-ﬂoor investors pay much higher transactions costs
than the on-ﬂoor market makers. Total costs of taking such a position may be prohibitive for offﬂoor investors. (2) Off-ﬂoor investors cannot really make a simultaneous decision to buy (sell)
the underlying, and buy or sell the implied puts or calls to construct the strategy. By the time these
strategies are communicated to the ﬂoor, prices could move. (3) Even if such opportunities are
found, net gains are often too small to make it worthwhile to take such positions sporadically. It
is, however, worthwhile to a market maker who specializes in these activities. (4) Finally, there
is also a serious risk associated with these positions, known as the pin risk.
2.2. A Remark on the Pin Risk
It is worthwhile to discuss the pin risk in more detail, since similar risks arise in hedging and
trading some exotic options as well. Suppose we put together a conversion at 100, and waited
90 days until expiration to unwind the position. The positions will expire some 90 days later
during a Friday. Suppose at expiration St is exactly 100. This means that the stock closes exactly
at the strike price. This leads to a dilemma for the market maker.
The market maker owns a stock. If he or she does not exercise the long put and if the short
call is not assigned (i.e., if he or she does not get to sell at K exactly), then the market maker
will have an open long position in the stock during the weekend. Prices may move by Monday
and he or she may experience signiﬁcant losses.
If, on the other hand, the market maker does exercise the long put (i.e., he or she sells the
stock at K) and if the call is assigned (i.e., he or she needs to deliver a stock at K), then the
market maker will have a short stock position during the weekend. These risks may not be great
for an end investor who takes such positions occasionally, but they may be substantial for a
professional trader who depends on these positions. There is no easy way out of this dilemma.
This type of risk is known as the pin risk.
The main cause of the pin risk is the kink in the expiration payoff at ST = K. A kink indicates
a sudden change in the slope—for a long call, from zero to one or vice versa. This means that even
with small movements in St , the hedge ratio can be either zero or one, and the market maker
may be caught signiﬁcantly off guard. If the slope of the payoff diagram changed smoothly,
then the required hedge would also change smoothly. Thus, a risk similar to pin risk may arise
whenever the delta of the instrument shows discrete jumps.
6
This is somewhat different from the upcoming strategy known as risk reversals.
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2.3. Risk Reversals
A more advanced version of the synthetic long and short futures positions is known as risk
reversals. These are liquid synthetics especially in the foreign exchange markets, where they
are traded as a commodity. Risk reversals are directional positions, but differ in more than one
way from synthetic long-short futures positions discussed in the previous section.
The idea is again to buy and sell calls and puts in order to replicate long and short futures
positions—but this time using options with different strike prices. Figure 10-6 shows an example.
The underlying is St . The strategy involves a short put struck at K1 , and a long call with strike K2 .
1
Long call
at expiration
ST
0
St
K2
2
1
K1
St
0
2
Short put
at expiration
Add together . . . This is a risk reversal
1
Long vol
ST
0
K1
K2
Risk-reversal at expiration
Short vol
2
FIGURE 10-6
ST
2. Option Strategies
287
Both options are out-of-the-money initially, and the St satisﬁes
K1 < S t < K 2
(13)
Since strikes can be chosen such that the put and call have the same premium, the risk reversal
can be constructed so as to have zero initial price.
By adding vertically the option payoffs in the top portion of Figure 10-6, we obtain the
expiration payoff shown at the bottom of the ﬁgure. If, at expiration, ST is between K1 and K2 ,
the strategy has zero payoff. If, at expiration, ST < K1 , the risk reversal loses money, but under
K2 < ST , it makes money. Clearly, what we have here is similar to a long position but the
position is neutral for small movements in the underlying starting from St . If taken naked, such
a position would imply a bullish view on St .
We consider an example from foreign exchange (FX) markets where risk reversals are traded
as commodities.
Example:
Twenty-ﬁve delta one-month risk reversals showed a stronger bias in favor of euro
calls (dollar puts) in the last two weeks after the euro started to strengthen against
the greenback.
Traders said market makers in EUR calls were buying risk reversals expecting further
euro upside. The one-month risk reversal jumped to 0.91 in favor of euro calls Wednesday from 0.3 three weeks ago. Implied volatility spiked across the board. One-month
volatility was 13.1% Wednesday from 11.78% three weeks ago as the euro appreciated
to USD1.0215 from USD1.0181 in the spot market.
The 25-delta risk reversals mentioned in this reading are shown in Figure 10-7a. The
risk reversal is constructed using two options, a call and a put. Both options are out-of-the-money
and have a “current” delta of 0.25. According to the reading, the 25-delta EUR call is more
expensive than the 25-delta EUR put.
2.3.1.
Uses of Risk Reversals
Risk reversals can be used as “cheap” hedging instruments. Here is an example.
Example:
A travel company in Paris last week entered a zero-cost risk reversal to hedge U.S. dollar
exposure to the USD. The company needs to buy dollars to pay suppliers in the U.S.,
China, Indonesia, and South America.
The head of treasury said it bought dollar calls and sold dollar puts in the transaction to hedge 30% of its USD200–300 million dollar exposure versus the USD. The
American-style options can be exercised between November and May.
The company entered a risk reversal rather than buying a dollar call outright because it
was cheaper. The head of treasury said the rest of its exposure is hedged using different
strategies, such as buying options outright. (Based on an article in Derivatives Week.)
Here we have a corporation that has EUR receivables from tourists going abroad but needs
to make payments to foreigners in dollars. Euros are received at time t, and dollars will be paid
at some future date T , with t < T . The risk reversal is put together as a zero cost structure,
which means that the premium collected from selling the put (on the USD) is equal to the call
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(a)
25-delta
long put
1
K1
ST
Tangent slope 5 2.25
St
2
(b)
1
25-delta
long call
K2
Tangent slope 5 .25
ST
2
(c)
Buy the put sell the call
for a 25-delta risk reversal. . .
1
K2
K1
St
ST
2
FIGURE 10-7
premium on the USD. For small movements in the exchange rate, the position is neutral, but for
large movements it represents a hedge similar to a futures contract.
Of course, such a position could also be taken in the futures market. But one important
advantage of the risk reversal is that it is “composed” of options, and hence involves, in general,
no daily mark-to-market adjustments.
2.4. Yield Enhancement Strategies
The class of option strategies that we have studied thus far is intended for creating synthetic
short and long futures positions. In this section, we consider option synthetics that are said to
lead to yield enhancement for investment portfolios.
2.4.1.
Call Overwriting
The simplest case is the following. At time t, an investor takes a long position in a stock with
current price St , as shown in Figure 10-8. If the stock price increases, the investor gains; if the